All of the 5th grade teachers and students from Loyola went on a field trip to an art museum. Tickets were $$6.50$ each for teachers and $$4.00$ each for students, and the group paid $$53.00$ in total. The next month, the same group visited a natural history museum where the tickets cost $$26.00$ each for teachers and $$7.50$ each for students, and the group paid $$127.00$ in total. Find the number of teachers and students on the field trips.
Answer: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${6.5x+4y = 53}$ ${26x+7.5y = 127}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-26x-16y = -212}$ ${26x+7.5y = 127}$ Add the top and bottom equations together. $ -8.5y = -85 $ $ y = \dfrac{-85}{-8.5}$ ${y = 10}$ Now that you know ${y = 10}$ , plug it back into $ {6.5x+4y = 53}$ to find $x$ ${6.5x + 4}{(10)}{= 53}$ $6.5x+40 = 53$ $6.5x = 13$ $x = \dfrac{13}{6.5}$ ${x = 2}$ You can also plug ${y = 10}$ into $ {26x+7.5y = 127}$ and get the same answer for $x$ ${26x + 7.5}{(10)}{= 127}$ ${x = 2}$ There were $2$ teachers and $10$ students on the field trips.